Finite Group Databasebeta

← #30770613

Subgroups — #30770613

Order 17152.

284 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 75 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22562464C211yesmaximal, characteristic
2#22562462C211yesmaximal, characteristic, complemented
3#22562438C211yesmaximal, characteristic, complemented
4#22559814C211yesmaximal, characteristic
5#22559807C211yesmaximal, characteristic, complemented
6#22558128C211yesmaximal, characteristic, complemented
7C67×#2240C211yesmaximal, characteristic
8C2×Dic536—22no—
9C2×#14523987—22no—
10#16863792—21no—
11#16863776C2211yescharacteristic
12#16863776C2211yescharacteristic
13#16863746—21no—
14#16863742—21no—
15#16863742—21no—
16#16863199—22no—
17C4×C67⋊C16C2211yescharacteristic
18C8×C67⋊C8C2211yescharacteristic
19#500×C67C2211yescharacteristic
20#493×C67C2211yescharacteristic
21#357×C67—22no—
22C4×C16×C67C2211yescharacteristic
23C2×Dic268—21no—
24C2×Dic268—21no—
25C2×Dic268—21no—
26C2×Dic268—21no—
27#14524043—21no—
28#14524043—21no—
29Dic536—44no—
30#14523987—44no—
31C2×C67⋊C16D412yes—
32#14523974—42no—
33#14523967—42no—
34#14523964—42no—
35#14523964—42no—
36C4×C67⋊C8D411yescharacteristic
37C4×C67⋊C8D411yescharacteristic
38C2×D8×C67—21no—
39C2×D8×C67—21no—
40C4⋊Q8×C67—21no—
41#127×C67—21no—
42C2×C16×C67D412yes—
43#107×C67—21no—
44#107×C67—21no—
45C4×C8×C67C2311yescharacteristic
46C2×Dic134—41no—
47C2×Dic134—41no—
48Dic268—42no—
49Dic268—42no—
50Dic268—42no—
51Dic268—42no—
52C2×C67⋊C8D812yes—
53C2×C67⋊C8D812yes—
54C4⋊Dic67—41no—
55C4⋊Dic67—41no—
56C4⋊Dic67—41no—
57C4⋊Dic67—41no—
58C67⋊C16—22no—
59C2×Q8×C67—41no—
60C2×D4×C67—41no—
61C2×D4×C67—41no—
62C2×D4×C67—41no—
63D8×C67—42no—
64D8×C67—42no—
65C2×C8×C67C2×D411yescharacteristic
66C2×C8×C67C2×D411yescharacteristic
67C4⋊C4×C67—41no—
68C4⋊C4×C67—41no—
69C42×C67C2×D411yescharacteristic
70C16×C67—22no—
71C23×C67—81no—
72Q8×C67—82no—
73D4×C67—81no—
74D4×C67—81no—
75D4×C67—82no—
76D4×C67—82no—
77C2×C4×C67—81no—
78C2×C4×C67#12711yescharacteristic
79C2×C4×C67C2×D811yescharacteristic
80C2×C4×C67C2×D811yescharacteristic
81C2×Dic67—81no—
82C2×Dic67—81no—
83Dic134—81no—
84Dic134—81no—
85Dic134—81no—
86Dic134—81no—
87C8×C67AGL(1,ℤ/8)12yes—
88C8×C67AGL(1,ℤ/8)12yes—
89C67⋊C8—22no—
90C67⋊C8—22no—
91C22×C67—161no—
92C22×C67—161no—
93C22×C67—161no—
94C22×C67#49311yescharacteristic
95C4×C67—161no—
96C4×C67#49611yescharacteristic
97C4×C67#49611yescharacteristic
98C4×C67#50812yes—
99C4×C67#51012yes—
100Dic67—161no—
101Dic67—161no—
102#12530—671nomaximal, Sylow
103C2×C67—321no—
104C2×C67#172111yescharacteristic
105C2×C67#224911yescharacteristic
106C2×C67#225111yescharacteristic
107#2248—671no—
108#2247—671no—
109#2241—671no—
110#2240D6711yescharacteristic
111#1709—671no—
112#1707—671no—
113#1312—671no—
114C67#1253011yescharacteristic, complemented, Sylow
115C2×Q32—1342no—
116C2×SD32—1342no—
117#500C2×D6711yescharacteristic
118#498—1341no—
119#494—671no—
120#494—671no—
121#493C2×D6711yescharacteristic
122#487—1341no—
123#486—1341no—
124#486—1341no—
125#358—1342no—
126#357—22no—
127C4×C16—671no—
128C4×C16C2×D6711yescharacteristic
129C82—671no—
130C2×Q16—1341no—
131C2×Q16—1341no—
132C2×Q16—1341no—
133C2×Q16—1341no—
134C2×D8—21no—
135C2×D8—21no—
136C4⋊Q8—21no—
137C4⋊Q8—1341no—
138C4⋊Q8—1341no—
139#127—21no—
140Q32—2684no—
141SD32—2684no—
142C2×C16—672no—
143C2×C16C22⋊D6712yes—
144#107—21no—
145#107—21no—
146#106—2682no—
147#103—2682no—
148#102—2682no—
149#102—2682no—
150C4×C8—671no—
151C4×C8—671no—
152C4×C8C22×D6711yescharacteristic
153C2×Q8—41no—
154C2×Q8—2681no—
155C2×Q8—2681no—
156C2×D4—41no—
157C2×D4—41no—
158C2×D4—41no—
159Q16—2682no—
160Q16—2682no—
161Q16—2682no—
162Q16—2682no—
163D8—42no—
164D8—42no—
165C2×C8—672no—
166C2×C8—672no—
167C2×C8D4×D6711yescharacteristic
168C2×C8C2×C22⋊D6711yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—2681no—
172C4⋊C4—2681no—
173C4⋊C4—2681no—
174C4⋊C4—2681no—
175C42D4×D6711yescharacteristic
176C16—22no—
177C16—1342no—
178C23—81no—
179Q8—82no—
180Q8—5361no—
181Q8—5361no—
182Q8—5361no—
183Q8—5361no—
184D4—81no—
185D4—81no—
186D4—82no—
187D4—82no—
188C2×C4—81no—
189C2×C4—5361no—
190C2×C4—5361no—
191C2×C4#1452403911yescharacteristic
192C2×C4D8×D6711yescharacteristic
193C2×C4D8×D6711yescharacteristic
194C8—1342no—
195C8—1342no—
196C8#1452406812yes—
197C8#1452407012yes—
198C22—161no—
199C22—161no—
200C22—161no—
201C22#1686377111yescharacteristic
202C4—161no—
203C4—10721no—
204C4—10721no—
205C4#1686378211yescharacteristic
206C4#1686378611yescharacteristic
207C4#1686382512yes—
208C4#1686383512yes—
209C2—321no—
210C2#2255987711yescharacteristic
211C2#2256246711yescharacteristic
212C2#2256247911yescharacteristic
subgroupG/H
centerC22#16863771
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#2240C2
socleC22×C67#493
prime radical#493×C67C22
hypercenter#2240D67
hypocenterC67#12530
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2240

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #22562462C2semidirect
3. #22562438C2semidirect
5. #22559807C2semidirect
6. #22558128C2semidirect
114. C67#12530semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#225624642
#225624622
#225624382
#225598142
#225598072
#225581282
C67×#22402
#1253067